I have a good handle on which verbs take être as an auxiliary, but why does être exist as an auxiliary? Is there some linguistic or historical rationale for existence of two auxiliary verbs in French? ...

Can someone point to some strict rules or rules of thumb for using être/avoir for passé composé for irregular verbs? Does it follow from change of radical for different tense?
In German, a change of ...

Please help me understand how does the main verb in passé composé change (especially when the subject is not singular). It would be helpful if you could detect the mistakes in the following sentences ...

I'm completely clear on the different uses of the two tenses, except for pretty much this one verb.
Can I get some different examples, with translations, of when you would say Il devait as opposed to ...